You can't drag a split-off segment of a clip into the media browser from a project. When you split or trim a clip, it only applies to the specific project timeline that you are working on. When you click on the Project Media item at the top of the project sidebar, you will see a display of all clips in the project, but they will be the full unedited clips. There is no way to add a split-off segment to the Project Media view or to the media browser in general.
What you can do is select the segment and do an Edit/Copy. Then open a newly created project and Edit/Paste the segment into it. Then share out the newly created project as a File to your desktop. Name it Interview Segment 1 (or whatever identifying title that you want).
Next in iMovie create a new event (select your library in the sidebar, do a File/New Event). Name the new event Segment 1, or whatever you want. Open the event. The media browser will display no media because there is no media in the event. Now import (drag) the shared movie into the empty media browser for the event. The edits and split that you made to the clip will be preserved as an independent clip in the event, that you can access from other projects. You can repeat this process for each segment and store each in a separate event or store them all in the same event. You can reference the title of each segment by selecting it in the media browser. The clip information will appear above the Preview screen.
Another way is to create a new project titled, for example, "Interview Segments", and paste all of your shared out segments (as described above) into it. The project will act as a library for your segments. You can copy and paste them into other projects.
As a note, instead of adding additional questions on a new subject to your initial post, it is more helpful to others if you would ask the question in a separate post. That way they can search and find it.
-- Rich